Wave analysis in generalized fractional Tzitzéica-type nonlinear PDEs: Contributions to nonlinear sciences

Alexandria Engineering Journal 92 (2024) 102–116 Available online 4 March 2024 1110-0168/© 2024 THE AUTHORS. Published by Elsevier BV on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Contents lists available at ScienceDirect Alexandria Engineering Journal journal homepage: www.elsevier.com/locate/aej Original Article Wave analysis in generalized fractional Tzitzéica-type nonlinear PDEs: Contributions to nonlinear sciences Naeem Ullah a, Hamood Ur Rehman b, Muhammad Imran Asjad a,∗, Muhammad Bilal Riaz c,d, Taseer Muhammad e aDepartment of Mathematics, University of Management and Technology, Lahore, Pakistan bDepartment of Mathematics, University of Okara, Okara, Pakistan cIT4Innovations, VSB–Technical University of Ostrava, Ostrava, Czech Republic dDepartment of Computer Science and Mathematics, Lebanese American University, Byblos, Lebanon eDepartment of Mathematics, College of Science, King Khalid University, Abha, Saudi Arabia A R T I C L E I N F O A B S T R A C T Keywords: Extended direct algebraic method Traveling wave structures Non-linear Tzitzéica type equations In this paper, the extended direct algebraic approach with the general fractional derivative is employed to attain various novel wave structures of the non-linear fractional Tzitzéica type non-linear evolution equations in the form of kink, singular, periodic singular, dark, bright and dark-bright combine solitons. For the purpose to illustrate the physical behavior of the acquired solutions, some of the extracted results are sketched in the pattern of 3-D and 2-D plots which show the efficiency and authenticity of the proposed method. The under consideration method can also utilize to any other non-linear model appears in optics and engineering. 1. Introduction Non-linear physical phenomena are dominant in natural life and have excited the curiosity of researchers for decades. In physics and engineering, developing the analytical or numerical solutions of fractional mathematical models for certain phenomena have become significant topics. For highly understanding the behavior of these complicated natural phenomena have mostly been established over these models. Recently, fractional operators are considered and mathematically dignified. The several properties of fractional operators have developed a great interest in fractional calculus in nowadays, also an extensive diversity of applications in the field of fluid dynamics, plasma physics, optical fiber, atomic science, engineering, mathematical biology, and several others [26–32,36,37]. During the previous 20 years, fractional calculus has become more popular and significant. Leibnitz wrote a letter to the hospital regarding the definition of a non-integer derivative, in this way a wide domain’s history opened. The ordinary differential equations (DEs) are changed into fractional DEs which are utilized in a diversity of mathematical modeling in different areas like as rheology, epidemiology and computer science. Non-locality shows an energetic role in numerous non-integer derivative models. Recently, many scholars have revealed that non-linear fractional differential equations (NFDEs) have a great impact in different arenas, as well as physics, engineering, biology, wave dynamics, control theory and many more. Fractional derivative is defined in different ways like, Caputo, Riemann-Liouville, Hadamard, Jumarie and Weyl that has been used positively in various fields, however all of these definitions have their benefits as well as drawbacks. The Riemann-Liouville derivative has deceptive more severe problem is that it is incapable to offer the derivative of a constant equal to zero. Moreover, if a function is a constant at the origin, its fractional derivative has a uniqueness at the origin, like as exponential and Mittag-Lefer functions. Because of these drawbacks, the range of applicability of Riemann-Liouville fractional derivatives is restricted. While the Caputo derivative is powerless to deal problems with a singular kernel. Due to these limits the scholars have stimulated to discover more appropriate definitions that have a fractional-order and are more comprehensive. A novel well-mannered simple non-integer derivative like the conformable derivative, was defined by Khalil et al. [25] depending on the derivative’s fundamental limit formulation. Due to distinction between Caputo formulations and Riemann-Liouville, the conformable derivative fulfils many * Corresponding author. E-mail address: [email protected] (M.I. Asjad). https://doi.org/10.1016/j.aej.2024.02.045 Received 3 December 2023; Received in revised form 6 February 2024; Accepted 22 February 2024 Alexandria Engineering Journal 92 (2024) 102–116 103 N. Ullah, H.U. Rehman, M.I. Asjad et al. crucial features. The author, established in Abdelhakim [3]that, for certain functions, the Caputo definition cannot yield longer results than the conformable definition in Khalil et al. [25]. In this study, our aim is to utilize a novel generalized definition of the fractional-order derivative that has benefits over other earlier definitions to achieve direct solutions of NFDEs. Because of wide applications in many research areas, NFDEs have become more charming and are rising gradually. Considering the essential part of analytical solutions of NFDEs in non-linear research, it would be valuable to explore possible novel soliton solutions to the non-linear fractional model. One of the most artistic developments in theoretic physics and non-linear science has been the establishment of methods for determining exact solutions for NFDEs. Various successful approaches have been established due to the rapid advancement in non-linear sciences for instance the solitary wave Ansatz method [8,9], the sine-Gordon expansion technique [16], Fan sub-equation approach [15], sine-cosine method [43], F-expansion scheme [45], modified Kudryashov scheme [10], modified simple equation scheme [38], [21–23]; [33]; [46]; [39]; [13]; [11,12]; [7]; [5,6]; [2]; [24]; [34]; [14] and so on. One of the most significant analytical approach for finding the exact solutions of nonlinear PDEs is the extended direct algebraic method (EDAM). This approach has been used successfully to construct many significant nonlinear models [1,8,9,19,35]. In mathematical modeling various fractional models are implemented by powerful numerical schemes for the analysis of many diseases dynamics, including HIV/AIDS, COVID-19, malaria, tuberculosis, also it is beneficial for controlling and monitoring the diseases [17,18]. The main advantage of the EDAM over other methods is that it offers more generalized solutions, which produce some known solutions by choosing appropriate parameters. The Tzitzéica-Dodd-Bullough-Mikhailov (TDBM), Tzitzéica-Dodd-Bullough (TDB), and Liouville non-linear equations that arise in optics. Tzitzéica’s study [42]in 1910 established the TDBM equation. Tzitzéica type equation is the improved form of TDBM and engaged in various researches for the previous decades, comprising in works ([44]and [41]). The fractional form of these equations is given like as: 𝑡𝐷𝐺𝐹𝐷 2𝛼𝑢−𝑥𝐷𝐺𝐹𝐷 2𝛽𝑢−𝑒𝑢+𝑒−2𝑢=0,(1) 𝑡𝐷𝐺𝐹𝐷 𝛼 𝑥𝐷𝐺𝐹𝐷 𝛽𝑢+𝑒𝑢+𝑒−2𝑢=0,(2) 𝑡𝐷𝐺𝐹𝐷 𝛼 𝑥𝐷𝐺𝐹𝐷 𝛽𝑢−𝑒𝑢−𝑒−2𝑢=0,(3) where 𝑡 >0and 0 <𝛼, 𝛽≤1. Several scientific demonstrations are created using these equations, such as dislocations in crystals, physics, nonlinear optics, and mechanics. The main objective of this work is to develop novel form of wave solutions to fractional Tzitzéica type NLEs using generalized fractional derivative. The fractional Tzitzeica-type nonlinear evolution equations appear in such problems in which fluid flow is varying to quantum theory. Furthermore, these equations participate in many arenas such as the circulation of fluxons in Josephson junctions between two superconductors, nonlinear optics, the motion of inflexible weights attached to a stretched wire, dislocations in metals and solid state physics. To the best of our current knowledge, the EDAM has not yet been employed to generalized fractional Tzitzeica-type nonlinear evolution equations to discover soliton solutions. The application of EDAM extends to several fields of non-linear sciences. However, this method is improved and applied on a non-linear fractional models. In this study, our primary focus is to establish advanced and widely applicable soliton solutions for generalized fractional Tzitzeica-type nonlinear evolution equations using the suggested method. The established solutions show wave-like behavior and are stated in trigonometric, exponential and hyperbolic forms. Furthermore, the soliton solutions attained from this study will also subsidize to the interpretation of complex phenomena associated with these specific fractional models. This article is structured as follows: In Sect. 2, some properties of generalized fractional derivative and narrative of extended direct algebraic method are presented. In Sect. 3, the solutions of fractional Tzitzéica type evolution equations have been acquired using the under discussion technique. In Sect. 4, graphical illustration of some selected solutions has been given. Lastly, findings of this study are given in Sect. 5. 2. The generalized fractional derivative This section deals with few basic definitions and concepts about the generalized fractional-derivative (GFD) [4]. Definition 1. If f:(0; ∞) ⟶ℜthen the GFD of 𝑓of order 0 <𝛼≤1is stated as 𝑡𝐷𝐺𝐹𝐷 𝛼𝑓(𝑡) = lim 𝜖→0 𝑓(𝑡+Γ(𝜚) Γ(𝜚−𝛼+1) 𝜖𝑡 1−𝛼)−𝑓(𝑡) 𝜖,𝜚>−1,𝜚∈ℜ+.(4) Some basic results and formulas of GFD are potted, as Theorem 1. If 𝛼∈(0, 1] and 𝑓, 𝑔be 𝛼-differentiable at a point, then ∙𝑡𝐷𝐺𝐹𝐷 𝛼(𝑎𝑓 +𝑏𝑔)=𝑎𝑡𝐷𝐺𝐹 𝐷 𝛼𝑓+𝑏𝑡𝐷𝐺𝐹𝐷 𝛼𝑔, ∀𝑎, 𝑏 ∈ℜ. ∙𝑡𝐷𝐺𝐹𝐷 𝛼(𝑡𝑣)=𝑣Γ(𝜚) Γ(𝜚−𝛼+1)𝑡𝑣−𝛼,∀𝑣>−1,𝑣∈ℜ. ∙𝑡𝐷𝐺𝐹𝐷 𝛼(𝑓𝑔)=𝑓𝑡𝐷𝐺𝐹𝐷 𝛼𝑔+𝑔𝑡𝐷𝐺𝐹𝐷 𝛼𝑓. ∙𝑡𝐷𝐺𝐹𝐷 𝛼(𝑓 𝑔)=𝑔𝑡𝐷𝐺𝐹𝐷 𝛼𝑓−𝑓𝑡𝐷𝐺𝐹𝐷 𝛼𝑔 𝑔2. If 𝑓is differentiable, then ∙𝑡𝐷𝐺𝐹 𝐷 𝛼𝑓(𝑡) =𝑣Γ(𝜚) Γ(𝜚−𝛼+1) 𝑡𝑣−𝛼𝑑𝑓 𝑑𝑡 . 2.1. Narrative of the method Assume the following NFPDE 𝐻(𝑢, 𝑡𝐷𝐺𝐹𝐷 𝛼𝑢, 𝑥𝐷𝐺𝐹𝐷 2𝛽𝑢, 𝑡𝐷𝐺𝐹𝐷 2𝛼𝑢, 𝑥𝐷𝐺𝐹𝐷 2𝛽𝑢, 𝑡𝐷𝐺𝐹𝐷 𝛼 𝑥𝐷𝐺𝐹𝐷 𝛽𝑢, ....)=0,(5) Alexandria Engineering Journal 92 (2024) 102–116 104 N. Ullah, H.U. Rehman, M.I. Asjad et al. where 0 <𝛼, 𝛽≤1and 𝑢 =𝑢(𝑥, 𝑡)is an unknown function, 𝐷𝛼 𝑡𝑢and 𝐷𝛽 𝑥𝑢are GFD of 𝑢and 𝐻is a polynomial in 𝑢and its derivative. The steps of extended direct algebraic method are elaborated as follows: Step 1: Pursue the wave variable of Eq. (5)to change FPDE into ODE, as 𝑢(𝑥, 𝑡)=𝑢(𝜉),𝜉=𝑘Γ(𝜚−𝛽+1) 𝛽Γ(𝜚)𝑥𝛽−𝐿Γ(𝜚−𝛼+1) 𝛼Γ(𝜚)𝑡𝛼,(6) where 𝑘and 𝐿are constants. Using Eq. (6) into Eq. (5)and yield a NODE as follows: 𝑅(𝑢, −𝐿𝑢′,𝑘𝑢 ′,𝐿 2𝑢′,𝑘 2𝑢′′,−𝑘𝐿𝑢′′, ...)=0,(7) here prime indicates the derivatives. Step 2: On integration Eq. (7)as per need and keep the integration constants equal to zero. An Eq. (7)adopt the solutions as 𝑢(𝜉)= 𝑁 ∑ 𝑖=0 𝑓𝑖Ψ𝑖(𝜉),(8) where 𝑓𝑖(0 ≤𝑖 ≤𝑁)are real numbers. Ψ′(𝜉)=𝐿𝑛(𝑃)(𝛿+ΛΨ(𝜉)+𝜎Ψ2(𝜉)),𝑃≠0,1.(9) The families of solutions of (9)are given, like as Family 1: If Λ2−4𝛿𝜎 < 0𝑎𝑛𝑑 𝜎 ≠0, Ψ1(𝜉)=−Λ 2𝜎+√−(Λ2−4𝛿𝜎) 2𝜎𝑡𝑎𝑛𝑃(√−(Λ2−4𝛿𝜎) 2𝜉), Ψ2(𝜉)=−Λ 2𝜎−√−(Λ2−4𝛿𝜎) 2𝜎𝑐𝑜𝑡𝑃(√−(Λ2−4𝛿𝜎) 2𝜉), Ψ3(𝜉)=−Λ 2𝜎+√−(Λ2−4𝛿𝜎) 2𝜎𝑡𝑎𝑛𝑃(√−(Λ2−4𝛿𝜎)𝜉)± √−𝑟𝑠(Λ2−4𝛿𝜎) 2𝜎𝑠𝑒𝑐𝑃(√−(Λ2−4𝛿𝜎)𝜉), Ψ4(𝜉)=−Λ 2𝜎−√−(Λ2−4𝛿𝜎) 2𝜎𝑐𝑜𝑡𝑃(√−(Λ2−4𝛿𝜎)𝜉)± √−𝑟𝑠(Λ2−4𝛿𝜎) 2𝜎𝑐𝑠𝑐𝑃(√−(Λ2−4𝛿𝜎)𝜉), Ψ5(𝜉)=−Λ 2𝜎+√−(Λ2−4𝛿𝜎) 4𝜎𝑡𝑎𝑛𝑃(√−(Λ2−4𝛿𝜎 4𝜉)−√−(Λ2−4𝛿𝜎) 4𝜎𝑐𝑜𝑡𝐴(√−(Λ2−4𝛿𝜎) 4𝜉). Family 2: If Λ2−4𝛿𝜎 > 0𝑎𝑛𝑑 𝜎 ≠0, Ψ6(𝜉)=−Λ 2𝜎−√(Λ2−4𝛿𝜎) 2𝜎𝑡𝑎𝑛ℎ𝑃(√(Λ2−4𝛿𝜎) 2𝜉), Ψ7(𝜉)=−Λ 2𝜎−√(Λ2−4𝛿𝜎) 2𝜎𝑐𝑜ℎ𝑡𝑃(√(Λ2−4𝛿𝜎) 2𝜉), Ψ8(𝜉)=−Λ 2𝜎−√(Λ2−4𝛿𝜎) 2𝜎𝑡𝑎𝑛ℎ𝑃(√Λ2−4𝛿𝜎𝜉)±𝜄√𝑟𝑠(Λ2−4𝛿𝜎) 2𝜎𝑠𝑒𝑐ℎ𝑃(√Λ2−4𝛿𝜎𝜉), Ψ9(𝜉)=−Λ 2𝜎−√(Λ2−4𝛿𝜎) 2𝜎𝑐𝑜𝑡ℎ𝑃(√Λ2−4𝛿𝜎𝜉)± √𝑟𝑠(Λ2−4𝛿𝜎) 2𝜎𝑐𝑠𝑐ℎ𝑃(√Λ2−4𝛿𝜎𝜉), Ψ10(𝜉)=−Λ 2𝜎−√(Λ2−4𝛿𝜎) 4𝜎𝑡𝑎𝑛ℎ𝑃(√Λ2−4𝛿𝜎 4𝜉)− √Λ2−4𝛿𝜎 4𝜎𝑐𝑜𝑡ℎ𝑃(√Λ2−4𝛿𝜎 4𝜉). Family 3: If 𝛿𝜎 > 0𝑎𝑛𝑑 Λ =0, Ψ11(𝜉)=√𝛿 𝜎𝑡𝑎𝑛𝑃(√𝛿𝜎𝜉), Ψ12(𝜉)=− √𝛿 𝜎𝑐𝑜𝑡𝑃(√𝛿𝜎𝜉), Ψ13(𝜉)=√𝛿 𝜎𝑡𝑎𝑛𝑃(2√𝛿𝜎𝜉)±√𝑟𝑠 𝛿 𝜎𝑠𝑒𝑐𝑃(2√𝛿𝜎𝜉), Ψ14(𝜉)=− √𝛿 𝜎𝑐𝑜𝑡𝑃(2√𝛿𝜎𝜉)±√𝑟𝑠 𝛿 𝜎𝑐𝑠𝑐𝑃(2√𝛿𝜎𝜉), Ψ15(𝜉)= 1 2(√𝛿 𝜎𝑡𝑎𝑛𝐴(√𝛿𝜎 2𝜉)−√𝛿 𝜎𝑐𝑜𝑡𝑃(√𝛿𝜎 2𝜉)). Family 4: If 𝛿𝜎 < 0𝑎𝑛𝑑 Λ =0, Alexandria Engineering Journal 92 (2024) 102–116 105 N. Ullah, H.U. Rehman, M.I. Asjad et al. Ψ16(𝜉)=− √−𝛿 𝜎𝑡𝑎𝑛ℎ𝑃(√−𝛿𝜎𝜉), Ψ17(𝜉)=− √−𝛿 𝜎𝑐𝑜𝑡ℎ𝑃(√−𝛿𝜎𝜉), Ψ18(𝜉)=− √−𝛿 𝜎𝑡𝑎𝑛ℎ𝑃(2√−𝛿𝜎𝜉)±𝜄√−𝑟𝑠 𝛿 𝜎𝑠𝑒𝑐ℎ𝑃(2√−𝛿𝜎𝜉), Ψ19(𝜉)=− √−𝛿 𝜎𝑐𝑜𝑡ℎ𝑃(2√−𝛿𝜎𝜉)±√−𝑟𝑠 𝛿 𝜎𝑐𝑠𝑐ℎ𝑃(2√−𝛿𝜎𝜉), Ψ20(𝜉)=−1 2(√−𝛿 𝜎𝑡𝑎𝑛ℎ𝑃(√−𝛿𝜎 2𝜉)+√−𝛿 𝜎𝑐𝑜𝑡ℎ𝑃(√−𝛿𝜎 2𝜉)). Family 5: If Λ =0𝑎𝑛𝑑 𝜎 =𝛿, Ψ21(𝜉)=𝑡𝑎𝑛𝑃(𝛿𝜉), Ψ22(𝜉)=−𝑐𝑜𝑡𝑃(𝛿𝜉), Ψ23(𝜉)=𝑡𝑎𝑛𝑃(2𝛿𝜉)±√𝑟𝑠𝑠𝑒𝑐 𝑃(2𝛿𝜉), Ψ24(𝜉)=−𝑐𝑜𝑡𝑃(2𝛿𝜉)±√𝑟𝑠𝑐𝑠𝑐 𝑃(2𝛿𝜉), Ψ25(𝜉)= 1 2(𝑡𝑎𝑛𝑃(𝛿 2𝜉)−𝑐𝑜𝑡𝑃(𝛿 2𝜉)). Family 6: If Λ =0𝑎𝑛𝑑 𝜎 =−𝛿, Ψ26(𝜉)=−𝑡𝑎𝑛ℎ𝑃(𝛿𝜉), Ψ27(𝜉)=−𝑐𝑜𝑡ℎ𝑃(𝛿𝜉), Ψ28(𝜉)=−𝑡𝑎𝑛ℎ𝑃(2𝛿𝜉)±𝜄√𝑟𝑠𝑠𝑒𝑐ℎ 𝑃(2𝛿𝜉), Ψ29(𝜉)=−𝑐𝑜𝑡ℎ𝐴(2𝛿𝜉)±√𝑟𝑠𝑐𝑠𝑐ℎ 𝑃(2𝛿𝜉), Ψ30(𝜉)=−1 2(𝑡𝑎𝑛ℎ𝑃(𝛿 2𝜉)+𝑐𝑜𝑡ℎ𝑃(𝛿 2𝜉)). Family 7: If Λ2=4𝛿𝜎, Ψ31(𝜉)= −2𝛿(Λ𝜉𝐿𝑛(𝑃)+2) Λ2𝜉𝐿𝑛(𝑃). Family 8: If Λ =𝑙, 𝛿 =𝑛𝑙 (𝑛 ≠0) 𝑎𝑛𝑑 𝜎 =0, Ψ32(𝜉)=𝑃𝑙𝜉 −𝑛. Family 9: If Λ =𝜎=0, Ψ33(𝜉)=𝛿𝜉𝐿𝑛(𝑃). Family 10: If Λ =𝛿=0, Ψ34(𝜉)= −1 𝜎𝜉𝐿𝑛(𝑃). Family 11: If 𝛿=0𝑎𝑛𝑑 Λ ≠0, Ψ35(𝜉)=− 𝑃Λ 𝜎(𝑐𝑜𝑠ℎ𝐴(Λ𝜉)−𝑠𝑖𝑛ℎ𝑃(Λ𝜉+𝑟), Ψ36(𝜉)=− Λ(𝑠𝑖𝑛ℎ𝑃(Λ𝜉)+𝑐𝑜𝑠ℎ𝑃(Λ𝜉)) 𝜎(𝑠𝑖𝑛ℎ𝑃(Λ𝜉)+𝑐𝑜𝑠ℎ𝑃(Λ𝜉)+𝑠). Family 12: If Λ =𝑙, 𝜎 =𝑛𝑙, (𝑛 ≠0) 𝑎𝑛𝑑 𝛿 =0, Ψ37(𝜉)= 𝑟𝑃 𝑙𝜉 𝑠−∓𝑃𝑙𝜉 . Note: The generalized hyperbolic and triangular functions are described like as [40] 𝑠𝑖𝑛ℎ𝑃(𝜉)=𝑟𝑃 𝜉−𝑠𝑃 −𝜉 2,𝑐𝑜𝑠ℎ 𝑃(𝜉)= 𝑟𝑃 𝜉+𝑠𝑃 −𝜉 2, 𝑡𝑎𝑛ℎ𝑃(𝜉)=𝑟𝑃 𝜉−𝑠𝑃 −𝜉 𝑟𝑃 𝜉+𝑠𝑃 −𝜉,𝑐𝑜𝑡ℎ 𝑃(𝜉)=𝑟𝑃 𝜉+𝑠𝑃 −𝜉 𝑟𝑃 𝜉−𝑠𝑃 −𝜉, 𝑠𝑒𝑐ℎ𝑃(𝜉)= 2 𝑟𝑃 𝜉+𝑠𝑃 −𝜉,𝑐𝑠𝑐ℎ 𝑃(𝜉)= 2 𝑟𝑃 𝜉−𝑠𝑃 −𝜉, 𝑠𝑖𝑛𝑃(𝜉)=𝑟𝑃 𝜄𝜉 −𝑠𝑃 −𝜄𝜉 2𝜄,𝑐𝑜𝑠 𝑃(𝜉)=𝑟𝑃 𝜄𝜉 +𝑠𝑃 −𝜄𝜉 2, Alexandria Engineering Journal 92 (2024) 102–116 106 N. Ullah, H.U. Rehman, M.I. Asjad et al. 𝑡𝑎𝑛𝑃(𝜉)=−𝜄𝑟𝑃 𝜄𝜉 −𝑠𝑃 −𝜄𝜉 𝑟𝑃 𝜄𝜉 +𝑠𝑃 −𝜄𝜉 ,𝑐𝑜𝑡 𝑃(𝜉)=𝜄𝑟𝑃 𝜄𝜉 +𝑠𝑃 −𝜄𝜉 𝑟𝑃 𝜄𝜉 −𝑠𝑃 −𝜄𝜉 , 𝑠𝑒𝑐𝑃(𝜉)= 2 𝑟𝑃 𝜄𝜉 +𝑠𝑃 −𝜄𝜉 ,𝑐𝑠𝑐 𝑃(𝜉)= 2𝜄 𝑟𝑃 𝜄𝜉 −𝑠𝑃 −𝜄𝜉 , where 𝑟, 𝑠 >0. Step 3: The value of 𝑁can be found by applying balance principle technique in (7). Substituting (6) into (5), gives a set of algebraic equations involving powers of Ψ𝑖(𝜉)(𝑖 =0, 1, 2, ...). Comparing the coefficients of Ψ(𝜉)to zero yields a bundle of equations. Step 4: On solutions the set of equations using Maple software, then insert the results into (9)to acquire the solutions of (1). 3. The Tzitzéica equation The fractional Tzitzéica equation is stated as 𝑡𝐷𝐺𝐹𝐷 2𝛼𝑢−𝑥𝐷𝐺𝐹𝐷 2𝛽𝑢−𝑒𝑢+𝑒−2𝑢=0,(10) where 0 <𝛼, 𝛽≤1and the parameters 𝛼and 𝛽indicating the fractional derivatives in time and space. Using the following wave transformations 𝑢(𝑥, 𝑡)=𝑢(𝜉),𝜉=𝑘Γ(𝜚−𝛽+1) 𝛽Γ(𝜚)𝑥𝛽−𝐿Γ(𝜚−𝛼+1) 𝛼Γ(𝜚)𝑡𝛼,(11) converts Eq. (10)as follows: (𝐿2−𝑘2)𝑢′′ −𝑒𝑢+𝑒−2𝑢=0,(12) where 𝑑𝑢 𝑑𝜉 =𝑢′. By choosing the transformation 𝑙𝑛𝑣 =𝑢or equivalently 𝑒𝑢=𝑣, Eq. (12), gives (𝐿2−𝑘2)(𝑣𝑣′′ −(𝑣′)2)−𝑣3+1=0.(13) 3.1. Applications of the new EDAM Using the balance principle on the terms 𝑣3and 𝑣𝑣′′, we get 𝑁=2. So, Eq. (8)converts to the following form 𝑢(𝜉)=𝑓0+𝑓1Ψ(𝜉)+𝑓2Ψ(𝜉)2,(14) where 𝑓0, 𝑓1and 𝑓2are constants. Plugging (14)in (13)and equating the arbitrary constants of Ψ(𝜉)to zero, a set of equations in Λ,𝜎,𝑓 0,𝑓 1,𝑓 2,𝐿 and 𝑘is achieved. On solution it, we get 𝑓0=−3 5,𝑓 1=4 5√𝜎 𝛿,𝑓 2=−8 5 𝜎 𝛿, 𝐿=√−1 5 −5𝛿𝜎𝐿𝑛(𝑃)2𝑘2+4 𝛿𝜎 𝐿𝑛(𝑃),𝑘=𝑘, Λ=−1 2√𝜎𝛿. (15) Eqns. (13), (14)and (15)give the families of solutions of (10), like as: Family 1: If Λ2−4𝛿𝜎 < 0and 𝜎≠0, then 𝑢1(𝑥, 𝑡)=ln[−3 5+4 5√𝜎 𝛿(−Λ 2𝜎+√−(Λ2−4𝛿𝜎) 2𝜎𝑡𝑎𝑛𝑃(√−(Λ2−4𝛿𝜎) 2𝜉)) −8 5 𝜎 𝛿(−Λ 2𝜎+√−(Λ2−4𝛿𝜎) 2𝜎𝑡𝑎𝑛𝑃(√−(Λ2−4𝛿𝜎) 2𝜉))2],(16) 𝑢2(𝑥, 𝑡)=ln[−3 5+4 5√𝜎 𝛿(−Λ 2𝜎−√−(Λ2−4𝛿𝜎) 2𝜎𝑐𝑜𝑡𝑃(√−(Λ2−4𝛿𝜎) 2𝜉)) −8 5 𝜎 𝛿(−Λ 2𝜎−√−(Λ2−4𝛿𝜎) 2𝜎𝑐𝑜𝑡𝑃(√−(Λ2−4𝛿𝜎) 2𝜉))2],(17) 𝑢3(𝑥, 𝑡)=ln[−3 5+4 5√𝜎 𝛿(−Λ 2𝜎+√−(Λ2−4𝛿𝜎) 2𝜎𝑡𝑎𝑛𝑃(√−(Λ2−4𝛿𝜎)𝜉) ±√−𝑟𝑠(Λ2−4𝛿𝜎) 2𝜎𝑠𝑒𝑐𝑃(√−(Λ2−4𝛿𝜎)𝜉)) −8 5 𝜎 𝛿(−Λ 2𝜎+√−(Λ2−4𝛿𝜎) 2𝜎𝑡𝑎𝑛𝑃(√−(Λ2−4𝛿𝜎)𝜉) ±√−𝑟𝑠(Λ2−4𝛿𝜎) 2𝜎𝑠𝑒𝑐𝑃(√−(Λ2−4𝛿𝜎)𝜉))2],(18) Alexandria Engineering Journal 92 (2024) 102–116 107 N. Ullah, H.U. Rehman, M.I. Asjad et al. 𝑢4(𝑥, 𝑡)=ln[−3 5+4 5√𝜎 𝛿(−Λ 2𝜎−√−(Λ2−4𝛿𝜎) 2𝜎𝑐𝑜𝑡𝑃(√−(Λ2−4𝛿𝜎)𝜉) ±√−𝑟𝑠(Λ2−4𝛿𝜎) 2𝜎𝑐𝑠𝑐𝑃(√−(Λ2−4𝛿𝜎)𝜉)) −8 5 𝜎 𝛿(−Λ 2𝜎−√−(Λ2−4𝛿𝜎) 2𝜎𝑐𝑜𝑡𝑃(√−(Λ2−4𝛿𝜎)𝜉) ±√−𝑟𝑠(Λ2−4𝛿𝜎) 2𝜎𝑐𝑠𝑐𝑃(√−(Λ2−4𝛿𝜎)𝜉))2],(19) 𝑢5(𝑥, 𝑡)=ln[−3 5+4 5√𝜎 𝛿(−Λ 2𝜎+√−(Λ2−4𝛿𝜎) 4𝜎𝑡𝑎𝑛𝑃(√−(Λ2−4𝛿𝜎) 4𝜉) −√−(Λ2−4𝛿𝜎) 4𝜎𝑐𝑜𝑡𝑃(√−(Λ2−4𝛿𝜎) 4𝜉)) −8 5 𝜎 𝛿(−Λ 2𝜎+√−(Λ2−4𝛿𝜎) 4𝜎𝑡𝑎𝑛𝑃(√−(Λ2−4𝛿𝜎) 4𝜉) −√−(Λ2−4𝛿𝜎) 4𝜎𝑐𝑜𝑡𝑃(√−(Λ2−4𝛿𝜎) 4𝜉))2].(20) Family 2: If Λ2−4𝛿𝜎 > 0. and 𝜎≠0, then 𝑢6(𝑥, 𝑡)=ln[−3 5+4 5√𝜎 𝛿(−Λ 2𝜎−√(Λ2−4𝛿𝜎) 2𝜎𝑡𝑎𝑛ℎ𝑃(√(Λ2−4𝛿𝜎) 2𝜉)) −8 5 𝜎 𝛿(−Λ 2𝜎−√(Λ2−4𝛿𝜎) 2𝜎𝑡𝑎𝑛ℎ𝑃(√(Λ2−4𝛿𝜎) 2𝜉))2],(21) 𝑢7(𝑥, 𝑡)=ln[−3 5+4 5√𝜎 𝛿(−Λ 2𝜎−√(Λ2−4𝛿𝜎) 2𝜎𝑐𝑜𝑡ℎ𝑃(√(Λ2−4𝛿𝜎) 2𝜉)) −8 5 𝜎 𝛿(−Λ 2𝜎−√(Λ2−4𝛿𝜎) 2𝜎𝑐𝑜𝑡ℎ𝑃(√(Λ2−4𝛿𝜎) 2𝜉))2],(22) 𝑢8(𝑥, 𝑡)=ln[−3 5+4 5√𝜎 𝛿(−Λ 2𝜎−√(Λ2−4𝛿𝜎) 2𝜎𝑡𝑎𝑛ℎ𝑃(√(Λ2−4𝛿𝜎)𝜉) ±𝜄√𝑟𝑠(Λ2−4𝛿𝜎) 2𝜎𝑠𝑒𝑐ℎ𝑃(√(Λ2−4𝛿𝜎)𝜉)) −8 5 𝜎 𝛿(−Λ 2𝜎−√(Λ2−4𝛿𝜎) 2𝜎𝑡𝑎𝑛ℎ𝑃(√(Λ2−4𝛿𝜎)𝜉) ±𝜄√𝑟𝑠(Λ2−4𝛿𝜎) 2𝜎𝑠𝑒𝑐ℎ𝑃(√(Λ2−4𝛿𝜎)𝜉))2],(23) 𝑢9(𝑥, 𝑡)=ln[−3 5+4 5√𝜎 𝛿(−Λ 2𝜎−√(Λ2−4𝛿𝜎) 2𝜎𝑐𝑜𝑡ℎ𝑃(√(Λ2−4𝛿𝜎)𝜉) ±√𝑟𝑠(Λ2−4𝛿𝜎) 2𝜎𝑐𝑠𝑐ℎ𝑃(√(Λ2−4𝛿𝜎)𝜉)) −8 5 𝜎 𝛿(−Λ 2𝜎−√(Λ2−4𝛿𝜎) 2𝜎𝑐𝑜𝑡ℎ𝑃(√(Λ2−4𝛿𝜎)𝜉) ±√𝑟𝑠(Λ2−4𝛿𝜎) 2𝜎𝑐𝑠𝑐ℎ𝑃(√(Λ2−4𝛿𝜎)𝜉))2],(24) 𝑢10(𝑥, 𝑡)=ln[−3 5+4 5√𝜎 𝛿(−Λ 2𝜎−√(Λ2−4𝛿𝜎) 4𝜎𝑡𝑎𝑛ℎ𝑃(√Λ2−4𝛿𝜎 4𝜉) −√(Λ2−4𝛿𝜎) 4𝜎𝑐𝑜𝑡ℎ𝑃(√(Λ2−4𝛿𝜎) 4𝜉)) −8 5 𝜎 𝛿(−Λ 2𝜎−√(Λ2−4𝛿𝜎) 4𝜎𝑡𝑎𝑛ℎ𝑃(√Λ2−4𝛿𝜎 4𝜉) Alexandria Engineering Journal 92 (2024) 102–116 108 N. Ullah, H.U. Rehman, M.I. Asjad et al. −√(Λ2−4𝛿𝜎) 4𝜎𝑐𝑜𝑡ℎ𝑃(√(Λ2−4𝛿𝜎) 4𝜉))2].(25) Family 3: If Λ2=4𝛿𝜎, then 𝑢11(𝑥, 𝑡)=ln[−3 5+4 5√𝜎 𝛿(−2 𝛿(Λ 𝜉𝐿𝑛(𝑃)+2 Λ2𝜉𝐿𝑛(𝑃))−8 5 𝜎 𝛿(−2 𝛿(Λ 𝜉𝐿𝑛(𝑃)+2 Λ2𝜉𝐿𝑛(𝑃))2].(26) 3.2. The DBM equation The fractional DBM equation is stated as 𝑡𝐷𝐺𝐹𝐷 𝛼 𝑥𝐷𝐺𝐹𝐷 𝛽𝑢+𝑒𝑢+𝑒−2𝑢=0.(27) Using the Eq. (11), we convert Eq. (27)to −𝐿𝑘𝑢′′ +𝑒𝑢+𝑒−2𝑢=0,(28) where 𝑑𝑢 𝑑𝜉 =𝑢′. By utilizing the transformation 𝑢 =𝑙𝑛𝑣 or equivalently 𝑣 =𝑒𝑢, Eq. (28), reduces to 𝑘𝐿(−𝑣𝑣′′ +(𝑣′)2)+𝑣3+1=0.(29) 3.3. Application of the new EDAM Here, we utilize the EDAM for the solutions of DBM equation. Utilizing balancing rule on (29), yields 𝑁=2, thus (8)converts to 𝑢(𝜉)=𝑓0+𝑓1Ψ(𝜉)+𝑓2Ψ(𝜉)2,(30) where 𝑓0, 𝑓1and 𝑓2are constants. Plugging (30)in (29)and equating the arbitrary constants of Ψ(𝜉)to zero, a system of equations in 𝑓0,𝑓 1,𝑓 2,Λ,𝑘, and 𝛿is achieved. On solution it, we get 𝑓0=1 2+1 2kL 𝐿𝑛(𝑃)2Λ2,𝑓 1=2kL Λ𝜎𝐿𝑛(𝑃)2, 𝑓2=2kL 𝜎2𝐿𝑛(𝑃)2,Λ=Λ ,𝐿=𝐿, 𝑘 =𝑘, 𝛿=3+kL 𝐿𝑛(𝑃)2Λ2 4kL 𝜎𝐿𝑛(𝑃)2.(31) Eqns. (29), (30)and (31)give the families of solutions of (28), like as: Family 1: If Λ2−4𝛿𝜎 < 0and 𝜎≠0, then 𝑢1(𝑥, 𝑡)=ln[1 2(1+kL 𝐿𝑛(𝑃)2(Λ2+4Λ𝜎(−Λ 2𝜎+√−(Λ2−4𝛿𝜎) 2𝜎𝑡𝑎𝑛𝑃(√−(Λ2−4𝛿𝜎) 2𝜉)) +4𝜎2(−Λ 2𝜎+√−(Λ2−4𝛿𝜎) 2𝜎𝑡𝑎𝑛𝑃(√−(Λ2−4𝛿𝜎) 2𝜉))2))],(32) 𝑢2(𝑥, 𝑡)=ln[1 2(1+kL 𝐿𝑛(𝑃)2(Λ2+4Λ𝜎(−Λ 2𝜎−√−(Λ2−4𝛿𝜎) 2𝜎𝑐𝑜𝑡𝑃(√−(Λ2−4𝛿𝜎) 2𝜉)) +4𝜎2(−Λ 2𝜎−√−(Λ2−4𝛿𝜎) 2𝜎𝑐𝑜𝑡𝑃(√−(Λ2−4𝛿𝜎) 2𝜉))2))],(33) 𝑢3(𝑥, 𝑡)=ln[1 2(1+kL 𝐿𝑛(𝑃)2(Λ2+4Λ𝜎(−Λ 2𝜎+√−(Λ2−4𝛿𝜎) 2𝜎𝑡𝑎𝑛𝑃(√−(Λ2−4𝛿𝜎)𝜉) ±√−𝑟𝑠(Λ2−4𝛿𝜎) 2𝜎𝑠𝑒𝑐𝑃(√−(Λ2−4𝛿𝜎)𝜉))+4𝜎2(−Λ 2𝜎+√−(Λ2−4𝛿𝜎) 2𝜎𝑡𝑎𝑛𝑃(√−(Λ2−4𝛿𝜎)𝜉) ±√−𝑟𝑠(Λ2−4𝛿𝜎) 2𝜎𝑠𝑒𝑐𝑃(√−(Λ2−4𝛿𝜎)𝜉))2))],(34) 𝑢4(𝑥, 𝑡)=ln[1 2(1+kL 𝐿𝑛(𝑃)2(Λ2+4Λ𝜎(−Λ 2𝜎−√−(Λ2−4𝛿𝜎) 2𝜎𝑐𝑜𝑡𝑃(√−(Λ2−4𝛿𝜎)𝜉) ±√−𝑟𝑠(Λ2−4𝛿𝜎) 2𝜎𝑐𝑠𝑐𝑃(√−(Λ2−4𝛿𝜎)𝜉))+4𝜎2(−Λ 2𝜎−√−(Λ2−4𝛿𝜎) 2𝜎𝑐𝑜𝑡𝑃(√−(Λ2−4𝛿𝜎)𝜉) ±√−𝑟𝑠(Λ2−4𝛿𝜎) 2𝜎𝑐𝑠𝑐𝑃(√−(Λ2−4𝛿𝜎)𝜉))2))],(35) Alexandria Engineering Journal 92 (2024) 102–116 109 N. Ullah, H.U. Rehman, M.I. Asjad et al. 𝑢5(𝑥, 𝑡)=ln[1 2(1+kL 𝐿𝑛(𝑃)2(Λ2+4Λ𝜎(−Λ 2𝜎+√−(Λ2−4𝛿𝜎) 4𝜎𝑡𝑎𝑛𝑃(√−(Λ2−4𝛿𝜎) 4𝜉) −√−(Λ2−4𝛿𝜎) 4𝜎𝑐𝑜𝑡𝑃(√−(Λ2−4𝛿𝜎) 4𝜉))+4𝜎2(−Λ 2𝜎+√−(Λ2−4𝛿𝜎) 4𝜎𝑡𝑎𝑛𝑃(√−(Λ2−4𝛿𝜎) 4𝜉) −√−(Λ2−4𝛿𝜎) 4𝜎𝑐𝑜𝑡𝑃(√−(Λ2−4𝛿𝜎) 4𝜉))2))].(36) Family 2: If Λ2−4𝛿𝜎 > 0. and 𝜎≠0, then 𝑢6(𝑥, 𝑡)=ln[1 2(1+kL 𝐿𝑛(𝑃)2(Λ2+4Λ𝜎(−Λ 2𝜎−√(Λ2−4𝛿𝜎) 2𝜎𝑡𝑎𝑛ℎ𝑃(√(Λ2−4𝛿𝜎) 2𝜉)) +4𝜎2(−Λ 2𝜎−√(Λ2−4𝛿𝜎) 2𝜎𝑡𝑎𝑛ℎ𝑃(√(Λ2−4𝛿𝜎) 2𝜉))2))],(37) 𝑢7(𝑥, 𝑡)=ln[1 2(1+kL 𝐿𝑛(𝑃)2(Λ2+4Λ𝜎(−Λ 2𝜎−√(Λ2−4𝛿𝜎) 2𝜎𝑐𝑜𝑡ℎ𝑃(√(Λ2−4𝛿𝜎) 2𝜉)) +4𝜎2(−Λ 2𝜎−√(Λ2−4𝛿𝜎) 2𝜎𝑐𝑜𝑡ℎ𝑃(√(Λ2−4𝛿𝜎) 2𝜉))2))],(38) 𝑢8(𝑥, 𝑡)=ln[1 2(1+kL 𝐿𝑛(𝑃)2(Λ2+4Λ𝜎(−Λ 2𝜎−√(Λ2−4𝛿𝜎) 2𝜎𝑡𝑎𝑛ℎ𝑃(√(Λ2−4𝛿𝜎)𝜉) ±𝜄√𝑟𝑠(Λ2−4𝛿𝜎) 2𝜎𝑠𝑒𝑐ℎ𝑃(√(Λ2−4𝛿𝜎)𝜉))+4𝜎2(−Λ 2𝜎−√(Λ2−4𝛿𝜎) 2𝜎𝑡𝑎𝑛ℎ𝑃(√(Λ2−4𝛿𝜎)𝜉) ±𝜄√𝑟𝑠(Λ2−4𝛿𝜎) 2𝜎𝑠𝑒𝑐ℎ𝑃(√(Λ2−4𝛿𝜎)𝜉))2))],(39) 𝑢9(𝑥, 𝑡)=ln[1 2(1+kL 𝐿𝑛(𝑃)2(Λ2+4Λ𝜎(−Λ 2𝜎−√(Λ2−4𝛿𝜎) 2𝜎𝑐𝑜𝑡ℎ𝑃(√(Λ2−4𝛿𝜎)𝜉) ±√𝑟𝑠(Λ2−4𝛿𝜎) 2𝜎𝑐𝑠𝑐ℎ𝑃(√(Λ2−4𝛿𝜎)𝜉))+4𝜎2(−Λ 2𝜎−√(Λ2−4𝛿𝜎) 2𝜎𝑐𝑜𝑡ℎ𝑃(√(Λ2−4𝛿𝜎)𝜉) ±√𝑟𝑠(Λ2−4𝛿𝜎) 2𝜎𝑐𝑠𝑐ℎ𝑃(√(Λ2−4𝛿𝜎)𝜉))2))],(40) 𝑢10(𝑥, 𝑡)=ln[1 2(1+kL 𝐿𝑛(𝑃)2(Λ2+4Λ𝜎(−Λ 2𝜎−√(Λ2−4𝛿𝜎) 4𝜎𝑡𝑎𝑛ℎ𝑃(√Λ2−4𝛿𝜎 4𝜉) −√(Λ2−4𝛿𝜎) 4𝜎𝑐𝑜𝑡ℎ𝑃(√(Λ2−4𝛿𝜎) 4𝜉))+4𝜎2(−Λ 2𝜎−√(Λ2−4𝛿𝜎) 4𝜎𝑡𝑎𝑛ℎ𝑃(√Λ2−4𝛿𝜎 4𝜉) −√(Λ2−4𝛿𝜎) 4𝜎𝑐𝑜𝑡ℎ𝑃(√(Λ2−4𝛿𝜎) 4𝜉))2))].(41) Family 3: If 𝛿𝜎 > 0𝑎𝑛𝑑 Λ =0, then 𝑢11(𝑥, 𝑡)=ln[1 2+2 kL 𝜎2𝐿𝑛(𝑃)2(−√𝛿 𝜎𝑡𝑎𝑛𝑃(√𝛿𝜎𝜉))2],(42) 𝑢12(𝑥, 𝑡)=ln[1 2+2 kL 𝜎2𝐿𝑛(𝑃)2(−√𝛿 𝜎𝑐𝑜𝑡𝑃(√𝛿𝜎𝜉))2],(43) 𝑢13(𝑥, 𝑡)=ln[1 2+2 kL 𝜎2𝐿𝑛(𝑃)2(√𝛿 𝜎𝑡𝑎𝑛𝑃(2√𝛿𝜎𝜉)±√𝑟𝑠 𝛿 𝜎𝑠𝑒𝑐𝑃(2√𝛿𝜎𝜉))2],(44) 𝑢14(𝑥, 𝑡)=ln[1 2+2 kL 𝜎2𝐿𝑛(𝑃)2(−√𝛿 𝜎𝑐𝑜𝑡𝑃(2√𝛿𝜎𝜉)±√𝑟𝑠 𝛿 𝜎𝑐𝑠𝑐𝑃(2√𝛿𝜎𝜉))2],(45) 𝑢15(𝑥, 𝑡)=ln[1 2+2 kL 𝜎2𝐿𝑛(𝑃)2(1 2(√𝛿 𝜎𝑡𝑎𝑛𝑃(√𝛿𝜎 2𝜉)−√𝛿 𝜎𝑐𝑜𝑡𝑃(√𝛿𝜎 2)𝜉))2].(46) Family 4: If 𝛿𝜎 < 0𝑎𝑛𝑑 Λ =0, then Alexandria Engineering Journal 92 (2024) 102–116 110 N. Ullah, H.U. Rehman, M.I. Asjad et al. 𝑢16(𝑥, 𝑡)=ln[1 2+2 kL 𝜎2𝐿𝑛(𝑃)2(−√−𝛿 𝜎𝑡𝑎𝑛ℎ𝑃(√−𝛿𝜎𝜉))2],(47) 𝑢17(𝑥, 𝑡)=ln[1 2+2 kL 𝜎2𝐿𝑛(𝑃)2(−√−𝛿 𝜎𝑐𝑜𝑡ℎ𝑃(√−𝛿𝜎𝜉))2],(48) 𝑢18(𝑥, 𝑡)=ln[1 2+2 kL 𝜎2𝐿𝑛(𝑃)2(−√−𝛿 𝜎𝑡𝑎𝑛ℎ𝑃(2√−𝛿𝜎𝜉)±𝜄√−𝑟𝑠 𝛿 𝜎𝑠𝑒𝑐ℎ𝑃(2√−𝛿𝜎𝜉))2],(49) 𝑢19(𝑥, 𝑡)=ln[1 2+2 kL 𝜎2𝐿𝑛(𝑃)2(−√−𝛿 𝜎𝑐𝑜𝑡ℎ𝑃(2√−𝛿𝜎𝜉 ±√−𝑟𝑠 𝛿 𝜎𝑐𝑠𝑐ℎ𝑃(2√−𝛿𝜎𝜉))2],(50) 𝑢20(𝑥, 𝑡)=ln[1 2+2 kL 𝜎2𝐿𝑛(𝑃)2(−1 2(√−𝛿 𝜎𝑡𝑎𝑛ℎ𝑃(√−𝛿𝜎 2𝜉)+√−𝛿 𝜎𝑐𝑜𝑡ℎ𝑃(√−𝛿𝜎 2𝜉)))2].(51) Family 5: If Λ =0𝑎𝑛𝑑 𝜎 =𝛿, then 𝑢21(𝑥, 𝑡)=ln[1 2+2 kL 𝜎2𝐿𝑛(𝑃)2(𝑡𝑎𝑛𝑃(𝛿𝜉))2],(52) 𝑢22(𝑥, 𝑡)=ln[1 2+2 kL 𝜎2𝐿𝑛(𝑃)2(−𝑐𝑜𝑡𝑃(𝛿𝜉))2],(53) 𝑢23(𝑥, 𝑡)=ln[1 2+2 kL 𝜎2𝐿𝑛(𝑃)2(𝑡𝑎𝑛𝑃(2𝛿𝜉)±√𝑟𝑠𝑠𝑒𝑐 𝑃(2𝛿𝜉))2],(54) 𝑢24(𝑥, 𝑡)=ln[1 2+2 kL 𝜎2𝐿𝑛(𝑃)2(−𝑐𝑜𝑡𝑃(2𝛿𝜉)±√𝑝𝑞 𝑐𝑠𝑐𝑃(2𝛿𝜉))2],(55) 𝑢25(𝑥, 𝑡)=ln[1 2+2 kL 𝜎2𝐿𝑛(𝑃)2(1 2(𝑡𝑎𝑛𝑃(𝛿 2𝜉)−𝑐𝑜𝑡𝑃(𝛿 2𝜉)))2].(56) Family 6: If Λ =0𝑎𝑛𝑑 𝜎 =−𝛿, then 𝑢26(𝑥, 𝑡)=ln[1 2+2 kL 𝜎2𝐿𝑛(𝑃)2(−𝑡𝑎𝑛ℎ𝑃(𝛿𝜉))2],(57) 𝑢27(𝑥, 𝑡)=ln[1 2+2 kL 𝜎2𝐿𝑛(𝑃)2(−𝑐𝑜𝑡ℎ𝑃(𝛿𝜉))2],(58) 𝑢28(𝑥, 𝑡)=ln[1 2+2 kL 𝜎2𝐿𝑛(𝑃)2(−𝑡𝑎𝑛ℎ𝑃(2𝛿𝜉)±𝜄√𝑟𝑠𝑠𝑒𝑐ℎ 𝑃(2𝛿𝜉))2],(59) 𝑢29(𝑥, 𝑡)=ln[1 2+2 kL 𝜎2𝐿𝑛(𝑃)2(−𝑐𝑜𝑡ℎ𝑃(2𝛿𝜉)±√𝑟𝑠𝑐𝑠𝑐ℎ 𝑃(2𝛿𝜉))2],(60) 𝑢30(𝑥, 𝑡)=ln[1 2+2 kL 𝜎2𝐿𝑛(𝑃)2(−1 2(𝑡𝑎𝑛ℎ𝑃(𝛿 2𝜉)+𝑐𝑜𝑡ℎ𝑃(𝛿 2𝜉)))2].(61) Family 7: If Λ2=4𝛿𝜎, then 𝑢31(𝑥, 𝑡)=ln[1 2(1+kL 𝐿𝑛(𝑃)2(Λ2+4Λ𝜎(−2 𝛿(Λ 𝜉𝐿𝑛(𝑃)+2 Λ2𝜉𝐿𝑛(𝑃)) +4𝜎2(−2 𝛿(Λ 𝜉𝐿𝑛(𝑃)+2 Λ2𝜉𝐿𝑛(𝑃))2))].(62) Family 8: If Λ =𝛿=0, then 𝑢32(𝑥, 𝑡)=ln[1 2+2 kL 𝜎2𝐿𝑛(𝑃)2(−1 𝜎𝜉𝐿𝑛(𝑃))2]. Family 9: If Λ =𝑙, 𝜎 =𝑛𝑙, (𝑛 ≠0) 𝑎𝑛𝑑 𝛿 =0, then 𝑢33(𝑥, 𝑡)=ln[1 2(1+kL 𝐿𝑛(𝑃)2(Λ2+4Λ𝜎(𝑟𝑃 𝑙𝜉 𝑠−∓𝑃𝑙𝜉 )+4𝜎2(𝑟𝑃 𝑙𝜉 𝑠−∓𝑃𝑙𝜉 )2))].(63)